This book consists of contributions from the participants of the Abel Symposium 2019 held in Alesund, Norway. It was centered about applications of the ideas of symmetry and invariance, including equivalence and deformation theory of geometric structures, classification of differential invariants and invariant differential operators, integrability analysis of equations of mathematical physics, progress in parabolic geometry and mathematical aspects of general relativity. The chapters are written by leading international researchers, and consist of both survey and research articles. The book gives the reader an insight into the current research in differential geometry and Lie theory, as well as applications of these topics, in particular to general relativity and string theory.

This brief approaches General Relativity from a planetary navigation perspective, delving into the unconventional mathematical methods required to produce computer software for space missions. It provides a derivation of the Einstein field equations and describes experiments performed on the Near Earth Asteroid Rendezvous mission, spanning General Relativity Theory from the fundamental assumptions to experimental verification. The software used for planetary missions is derived from mathematics that use matrix notation. An alternative is to use Einstein summation notation, which enables the mathematics to be presented in a compact form but makes the geometry difficult to understand. In this book, the relationship of matrix notation to summation notation is shown. The purpose is to enable the reader to derive the mathematics used in the software in either matrix notation or summation notation. This brief is a useful tool for advanced students and young professionals embarking on careers in planetary navigation.

"Wald's book is clearly the first textbook on general relativity with a totally modern point of view; and it succeeds very well where others are only partially successful. The book includes full discussions of many problems of current interest which are not treated in any extant book, and all these matters are considered with perception and understanding."--S. Chandrasekhar
"A "tour de force" lucid, straightforward, mathematically rigorous, exacting in the analysis of the theory in its physical aspect."--L. P. Hughston, "Times Higher Education Supplement"
"Truly excellent. . . . A sophisticated text of manageable size that will probably be read by every student of relativity, astrophysics, and field theory for years to come."--James W. York, "Physics Today"

CHOICE Highly Recommended Title, August 2019 Expertly guided by renowned cosmologist Dr. David Lyth, learn about the pioneering scientists whose work provided the foundation for Einstein's formulation of his theories of relativity, and about Einstein's groundbreaking life and work as well. This highly readable and accessible panorama of the field delicately balances history and science as it takes the reader on an adventure through the centuries. Without complex mathematics or scientific formulae, this book will be of interest to all, even those without a scientific background, who are intrigued to find out more about what paved the way for one of our most famous physicists to push the boundaries of physics to new lengths. Features: Written by an internationally renowned physicist and cosmologist Describes the life and times of Einstein and his important predecessors Focuses on one of the most famous areas of science, Einstein's Relativity Theory

A groundbreaking textbook on twenty-first-century general relativity and cosmology Kip Thorne and Roger Blandford's monumental Modern Classical Physics is now available in five stand-alone volumes that make ideal textbooks for individual graduate or advanced undergraduate courses on statistical physics; optics; elasticity and fluid dynamics; plasma physics; and relativity and cosmology. Each volume teaches the fundamental concepts, emphasizes modern, real-world applications, and gives students a physical and intuitive understanding of the subject. Relativity and Cosmology is an essential introduction to the subject, including remarkable recent advances. Written by award-winning physicists who have made fundamental contributions to the field and taught it for decades, the book differs from most others on the subject in important ways. It highlights recent transformations in our understanding of black holes, gravitational waves, and the cosmos; it emphasizes the physical interpretation of general relativity in terms of measurements made by observers; it explains the physics of the Riemann tensor in terms of tidal forces, differential frame dragging, and associated field lines; it presents an astrophysically oriented description of spinning black holes; it gives a detailed analysis of an incoming gravitational wave's interaction with a detector such as LIGO; and it provides a comprehensive, in-depth account of the universe's evolution, from its earliest moments to the present. While the book is designed to be used for a one-quarter or full-semester course, it goes deep enough to provide a foundation for understanding and participating in some areas of cutting-edge research. Includes many exercise problems Features color figures, suggestions for further reading, extensive cross-references, and a detailed index Optional "Track 2" sections make this an ideal book for a one-quarter or one-semester course An online illustration package is available to professors The five volumes, which are available individually as paperbacks and ebooks, are Statistical Physics; Optics; Elasticity and Fluid Dynamics; Plasma Physics; and Relativity and Cosmology.

Ross P. Cameron argues that the flow of time is a genuine feature of reality. He suggests that the best version of the A-Theory is a version of the Moving Spotlight view, according to which past and future beings are real, but there is nonetheless an objectively privileged present. Cameron argues that the Moving Spotlight theory should be viewed as having more in common with Presentism (the view that reality is limited to the present) than with the B-Theory (the view that time is just another dimension like space through which things are spread out). The Moving Spotlight view, on this picture, agrees with Presentism that everything is the way it is now, it simply thinks that non-present beings are amongst the things that are now some way. Cameron argues that the Moving Spotlight theory provides the best account of truthmakers for claims about what was or will be the case, and he defends the view against a number of objections, including McTaggart's argument that the A-Theory is inconsistent, and the charge that if the A-Theory is true but presentism false then we could not know that we are present. The Moving Spotlight defends an account of the open future-that what will happen is, as yet, undetermined-Land argues that this is a better account than that available to the Growing Block theory.

Time magazine's "Man of the Century", Albert Einstein is the founder of modern physics and his theory of relativity is the most important scientific idea of the modern era. In this short book, Einstein explains, using the minimum of mathematical terms, the basic ideas and principles of the theory that has shaped the world we live in today. Unsurpassed by any subsequent books on relativity, this remains the most popular and useful exposition of Einstein's immense contribution to human knowledge. With a new foreword by Derek Raine.

This textbook on the nature of space and time explains the new theory of Space Dynamics, which describes the dynamics of gravity as the evolution of conformal 3-dimensional geometry. Shape Dynamics is equivalent to Einstein's General Relativity in those situations in which the latter has been tested experimentally, but the theory is based on different first principles. It differs from General Relativity in certain extreme conditions. Shape Dynamics allows us to describe situations in which the spacetime picture is no longer adequate, such as in the presence of singularities, when the idealization of infinitesimal rods measuring scales and infinitesimal clocks measuring proper time fails. This tutorial book contains both a quick introduction for readers curious about Shape Dynamics, and a detailed walk-through of the historical and conceptual motivations for the theory, its logical development from first principles and a description of its present status. It includes an explanation of the origin of the theory, starting from problems posed first by Newton more than 300 years ago. The book will interest scientists from a large community including all foundational fields of physics, from quantum gravity to cosmology and quantum foundations, as well as researchers interested in foundations. The tutorial is sufficiently self-contained for students with some basic background in Lagrangian/Hamiltonian mechanics and General Relativity.

The notion that fundamental equations governing the motions of physical systems are invariant under the time reversal transformation (T) has been an important, but often subliminal, element in the development of theoretical physics. It serves as a powerful and useful tool in analyzing the structure of matter at all scales, from gases and condensed matter to subnuclear physics and the quantum theory of fields. The assumption of invariance under T was called into question, however, by the 1964 discovery that a closely related assumption, that of CP invariance (where C is charge conjugation and P is space inversion), is violated in the decay of neutral K mesons.
In The Physics of Time Reversal, Robert G. Sachs comprehensively treats the role of the transformation T, both as a tool for analyzing the structure of matter and as a field of fundamental research relating to CP violation. For this purpose he reformulates the definitions of T, P, and C so as to avoid subliminal assumptions of invariance. He summarizes the standard phenomenology of CP violation in the K-meson system and addresses the question of the mysterious origin of CP violation. Using simple examples based on the standard quark model, Sachs summarizes and illustrates how these phenomenological methods can be extended to analysis of future experiments on heavy mesons. He notes that his reformulated approach to conventional quantum field theory leads to new questions about the meaning of the transformations in the context of recent theoretical developments such as non-Abelian gauge theories, and he suggests ways in which these questions may lead to new directions of research.

The book attempts to provide an introduction to quantum field theory emphasizing conceptual issues frequently neglected in more "utilitarian" treatments of the subject. The book is divided into four parts, entitled respectively "Origins", "Dynamics", "Symmetries", and "Scales". The emphasis is conceptual - the aim is to build the theory up systematically from some clearly stated foundational concepts - and therefore to a large extent anti-historical, but two historical Chapters ("Origins") are included to situate quantum field theory in the larger context of modern physical theories. The three remaining sections of the book follow a step by step reconstruction of this framework beginning with just a few basic assumptions: relativistic invariance, the basic principles of quantum mechanics, and the prohibition of physical action at a distance embodied in the clustering principle. The "Dynamics" section of the book lays out the basic structure of quantum field theory arising from the sequential insertion of quantum-mechanical, relativistic and locality constraints. The central role of symmetries in relativistic quantum field theories is explored in the third section of the book, while in the final section, entitled "Scales", we explore in detail the feature of quantum field theories most critical for their enormous phenomenological success - the scale separation property embodied by the renormalization group properties of a theory defined by an effective local Lagrangian.

Ever since its discovery eighty-five years ago, quantum theory has been used to study the physical universe with great profit, both intellectual and financial. Over the last fifty years, however, we have found out more and more about the theory itself, and what it tells us about the universe. It seems we may have to accept non-locality - cause and effect may be light-years apart; loss of realism - nature may be fundamentally probabilistic; and non-determinism - it seems that God does play dice! This book, written by an expert in the field, explains the emergence of our new perspective on quantum theory, but also describes how the ideas involved in this re-evaluation led seamlessly to a totally new discipline -quantum information theory. This discipline includes quantum computation, which is able to perform tasks quite out of the range of other computers; the totally secure algorithms of quantum cryptography; and quantum teleportation - as part of science fact rather than science fiction. The book is the first to combine these elements, and will be of interest to anybody interested in fundamental aspects of science and their application to the real world.

This book provides an accessible, yet thorough, introduction to special and general relativity, crafted and class-tested over many years of teaching. Suitable for advanced undergraduate and graduate students, this book provides clear descriptions of how to approach the mathematics and physics involved. It is also contains the latest exciting developments in the field, including dark energy, gravitational waves, and frame dragging. The table of contents has been carefully developed in consultation with a large number of instructors teaching courses worldwide, to ensure its wide applicability to modules on relativity and gravitation. Features: A clear, accessible writing style, presenting a sophisticated approach to the subject, that remains suitable for advanced undergraduate students and above Class-tested over many years To be accompanied by a partner volume on 'Advanced Topics' for students to further extend their learning

General Relativity is a beautiful geometric theory, simple in its mathematical formulation but leading to numerous consequences with striking physical interpretations: gravitational waves, black holes, cosmological models, and so on. This introductory textbook is written for mathematics students interested in physics and physics students interested in exact mathematical formulations (or for anyone with a scientific mind who is curious to know more of the world we live in), recent remarkable experimental and observational results which confirm the theory are clearly described and no specialised physics knowledge is required. The mathematical level of Part A is aimed at undergraduate students and could be the basis for a course on General Relativity. Part B is more advanced, but still does not require sophisticated mathematics. Based on Yvonne Choquet-Bruhat's more advanced text, General Relativity and the Einstein Equations, the aim of this book is to give with precision, but as simply as possible, the foundations and main consequences of General Relativity. The first five chapters from General Relativity and the Einstein Equations have been updated with new sections and chapters on black holes, gravitational waves, singularities, and the Reissner-Nordstroem and interior Schwarzchild solutions. The rigour behind this book will provide readers with the perfect preparation to follow the great mathematical progress in the actual development, as well as the ability to model, the latest astrophysical and cosmological observations. The book presents basic General Relativity and provides a basis for understanding and using the fundamental theory.

The book unifies quantum theory and the general theory of relativity. As an unsolved problem for about 100 years and influencing so many fields, this is probably of some importance to the scientific community. Examples like Higgs field, limit to classical Dirac and Klein-Gordon or Schroedinger cases, quantized Schwarzschild, Kerr, Kerr-Newman objects, and the photon are considered for illustration. An interesting explanation for the asymmetry of matter and antimatter in the early universe was found while quantizing the Schwarzschild metric.

Time is an illusion. Although the laws of physics create a powerful impression that time is flowing, in fact there are only timeless 'nows'. In THE END OF TIME, the British theoretical physicist Julian Barbour describes the coming revolution in our understanding of the world: a quantum theory of the universe that brings together Einstein's general theory of relativity, which denies the existence of a unique time, and quantum mechanics, which demands one. Barbour believes that only the most radical of ideas can resolve the conflict between these two theories: that there is, quite literally, no time at all. This is the first full-length account of the crisis in our understanding that has enveloped quantum cosmology. Unifying thinking that has never been brought together before in a book for the general reader, Barbour reveals the true architecture of the universe and demonstrates how physics is coming up sharp against the extraordinary possibility that the sense of time passing emerges from a universe that is timeless. The heart of the book is the author's lucid description of how a world of stillness can appear to be teeming with motion: in this timeless world where all possible instants coexist, complex mathematical rules of quantum mechanics bind together a special selection of these instants in a coherent order that consciousness perceives as the flow of time. Finally, in a lucid and eloquent epilogue, the author speculates on the philosophical implications of his theory: Does free will exist? Is time travel possible? How did the universe begin? Where is heaven? Does the denial of time make life meaningless? Written with exceptional clarity and elegance, this profound and original work presents a dazzlingly powerful argument that all will be able to follow, but no-one with an interest in the workings of the universe will be able to ignore.

This book is aimed at students making the transition from a first course on general relativity to a specialized subfield. It presents a variety of topics under the general headings of gravitational waves in vacuo and in a cosmological setting, equations of motion, and black holes, all having a clear physical relevance and a strong emphasis on space-time geometry. Each chapter could be used as a basis for an early postgraduate project for those who are exploring avenues into research in general relativity and who have already accumulated the required technical knowledge. The presentation of each chapter is research monograph style, rather than text book style, in order to impress on interested students the need to present their research in a clear and concise format. Students with advanced preparation in general relativity theory might find a treasure trove here.

Many regard Albert Einstein as the greatest physicist since Newton. What exactly did he do that is so important in physics? We provide an introduction to his physics at a level accessible to an undergraduate physics student. All equations are worked out in detail from the beginning. Einstein's doctoral thesis and his Brownian motion paper were decisive contributions to our understanding of matter as composed of molecules and atoms. Einstein was one of the founding fathers of quantum theory: his photon proposal through the investigation of blackbody radiation, his quantum theory of photoelectric effect and specific heat, his calculation of radiation fluctuation giving the first statement of wave-particle duality, his introduction of probability in the description of quantum radiative transitions, and finally the quantum statistics and Bose-Einstein condensation. Einstein's special theory of relativity gave us the famous E=mc(2) relation and the new kinematics leading to the idea of the 4-dimensional spacetime as the arena in which physical events take place. Einstein's geometric theory of gravity, general relativity, extends Newton's theory to time-dependent and strong gravitational fields. It laid the ground work for the study of black holes and cosmology. This is a physics book with material presented in the historical context. We do not stop at Einstein's discovery, but carry the discussion onto some of the later advances: Bell's theorem, quantum field theory, gauge theories and Kaluza-Klein unification in a spacetime with an extra spatial dimension. Accessibility of the material to a modern-day reader is the goal of our presentation. Although the book is written with primarily a physics readership in mind (it can also function as a textbook), enough pedagogical support material is provided that anyone with a solid background in introductory physics can, with some effort, understand a good part of this presentation.

"Flatland" is a unique, delightful satire that has charmed readers for over a century. Published in 1884 by the English clergyman and headmaster Edwin A. Abbott, it is the fanciful tale of A. Square, a two-dimensional being who is whisked away by a mysterious visitor to The Land of Three Dimensions, an experience that forever alters his worldview.

Like the original, Ian Stewart's commentary takes readers on a strange and wonderful journey. With clarity and wit, Stewart illuminates Abbott's numerous Victorian references and touches on such diverse topics as ancient Babylon, Karl Marx, Mary Shelley's "Frankenstein," Mt. Everest, H.G. Wells, and phrenology. "The Annotated Flatland" makes fascinating connections between "Flatland" and Abbott's era, resulting in a classic to rival Abbott's own, and a book that will inspire and delight curious readers for generations to come.

Relativity Made Relatively Easy presents an extensive study of Special Relativity and a gentle (but exact) introduction to General Relativity for undergraduate students of physics. Assuming almost no prior knowledge, it allows the student to handle all the Relativity needed for a university course, with explanations as simple, thorough, and engaging as possible. The aim is to make manageable what would otherwise be regarded as hard; to make derivations as simple as possible and physical ideas as transparent as possible. Lorentz invariants and four-vectors are introduced early on, but tensor notation is postponed until needed. In addition to the more basic ideas such as Doppler effect and collisions, the text introduces more advanced material such as radiation from accelerating charges, Lagrangian methods, the stress-energy tensor, and introductory General Relativity, including Gaussian curvature, the Schwarzschild solution, gravitational lensing, and black holes. A second volume will extend the treatment of General Relativity somewhat more thoroughly, and also introduce Cosmology, spinors, and some field theory.

Hyperbolic Dynamics and Brownian Motion illustrates the interplay between distinct domains of mathematics. There is no assumption that the reader is a specialist in any of these domains: only basic knowledge of linear algebra, calculus and probability theory is required. The content can be summarized in three ways: Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group. The Lorentz group plays, in relativistic space-time, a role analogue to the rotations in Euclidean space. The hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. Hyperbolic geometry is presented via special relativity to benefit from the physical intuition. Secondly, this book introduces basic notions of stochastic analysis: the Wiener process, Ito's stochastic integral, and calculus. This introduction allows study in linear stochastic differential equations on groups of matrices. In this way the spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and the relativistic diffusion are constructed. Thirdly, quotients of the hyperbolic space under a discrete group of isometries are introduced. In this framework some elements of hyperbolic dynamics are presented, as the ergodicity of the geodesic and horocyclic flows. This book culminates with an analysis of the chaotic behaviour of the geodesic flow, performed using stochastic analysis methods. This main result is known as Sinai's central limit theorem.

This book introduces the modern field of 3+1 numerical relativity. The book has been written in a way as to be as self-contained as possible, and only assumes a basic knowledge of special relativity. Starting from a brief introduction to general relativity, it discusses the different concepts and tools necessary for the fully consistent numerical simulation of relativistic astrophysical systems, with strong and dynamical gravitational fields. Among the topics discussed in detail are the following: the initial data problem, hyperbolic reductions of the field equations, gauge conditions, the evolution of black hole space-times, relativistic hydrodynamics, gravitational wave extraction and numerical methods. There is also a final chapter with examples of some simple numerical space-times. The book is aimed at both graduate students and researchers in physics and astrophysics, and at those interested in relativistic astrophysics.